
from __future__ import annotations

def a__       (			A_,	A_ = None,	A_ = None,	A_ = False,	):






  '''simple docstring'''





  __magic_name__								=			cipher_alphabet or [chr(A_   ) for i in range(97,	123   )]

  # If the argument is None or the user provided an empty dictionary
  if not frequencies_dict:
    # Frequencies of letters in the english language (how much they show up)
    __magic_name__								=			{
        """a""": 0.08497,
        """b""": 0.01492,
        """c""": 0.02202,
        """d""": 0.04253,
        """e""": 0.11162,
        """f""": 0.02228,
        """g""": 0.02015,
        """h""": 0.06094,
        """i""": 0.07546,
        """j""": 0.00153,
        """k""": 0.01292,
        """l""": 0.04025,
        """m""": 0.02406,
        """n""": 0.06749,
        """o""": 0.07507,
        """p""": 0.01929,
        """q""": 0.00095,
        """r""": 0.07587,
        """s""": 0.06327,
        """t""": 0.09356,
        """u""": 0.02758,
        """v""": 0.00978,
        """w""": 0.02560,
        """x""": 0.00150,
        """y""": 0.01994,
        """z""": 0.00077,
    }
  else:
    # Custom frequencies dictionary
    __magic_name__								=			frequencies_dict

  if not case_sensitive:
    __magic_name__								=			ciphertext.lower()

  # Chi squared statistic values
  __magic_name__								=			{}

  # cycle through all of the shifts
  for shift in range(len(A_   )   ):
    __magic_name__								=			""""""

    # decrypt the message with the shift
    for letter in ciphertext:
      try:
        # Try to index the letter in the alphabet
        __magic_name__								=			(alphabet_letters.index(letter.lower()   ) - shift) % len(
            A_   )
        decrypted_with_shift += (
            alphabet_letters[new_key].upper()
            if case_sensitive and letter.isupper()
            else alphabet_letters[new_key]
        )
      except ValueError:
        # Append the character if it isn't in the alphabet
        decrypted_with_shift += letter

    __magic_name__								=			0.0

    # Loop through each letter in the decoded message with the shift
    for letter in decrypted_with_shift:
      if case_sensitive:
        __magic_name__								=			letter.lower()
        if letter in frequencies:
          # Get the amount of times the letter occurs in the message
          __magic_name__								=			decrypted_with_shift.lower().count(A_   )

          # Get the excepcted amount of times the letter should appear based
          # on letter frequencies
          __magic_name__								=			frequencies[letter] * occurrences

          # Complete the chi squared statistic formula
          __magic_name__								=			((occurrences - expected) ** 2) / expected

          # Add the margin of error to the total chi squared statistic
          chi_squared_statistic += chi_letter_value
      else:
        if letter.lower() in frequencies:
          # Get the amount of times the letter occurs in the message
          __magic_name__								=			decrypted_with_shift.count(A_   )

          # Get the excepcted amount of times the letter should appear based
          # on letter frequencies
          __magic_name__								=			frequencies[letter] * occurrences

          # Complete the chi squared statistic formula
          __magic_name__								=			((occurrences - expected) ** 2) / expected

          # Add the margin of error to the total chi squared statistic
          chi_squared_statistic += chi_letter_value

        # Add the data to the chi_squared_statistic_values dictionary
    __magic_name__								=			(
        chi_squared_statistic,
        decrypted_with_shift,
    )

  # Get the most likely cipher by finding the cipher with the smallest chi squared
  # statistic
  def chi_squared_statistic_values_sorting_key(A_   ) -> tuple[float, str]:
    return chi_squared_statistic_values[key]

  __magic_name__								=			min(
      A_,	key=A_,	)

  # Get all the data from the most likely cipher (key, decoded message)
  (
      (
      __magic_name__
  )	,       (
      __magic_name__
  )	,       
  )								=			chi_squared_statistic_values[most_likely_cipher]

  # Return the data on the most likely shift
  return (
      most_likely_cipher,
      most_likely_cipher_chi_squared_value,
      decoded_most_likely_cipher,
  )



import math

def a__       (			A_   ):






  '''simple docstring'''





  if 1 < number < 4:
    # 2 and 3 are primes
    return True
  elif number < 2 or number % 2 == 0 or number % 3 == 0:
    # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
    return False

  # All primes number are in format of 6k +/- 1
  for i in range(5,	int(math.sqrt(A_   ) + 1   ),	6   ):
    if number % i == 0 or number % (i + 2) == 0:
      return False
  return True

def a__       (			A_ = 0.1   ):






  '''simple docstring'''





  __magic_name__								=			3
  __magic_name__								=			3

  while primes / (2 * j - 1) >= ratio:
    for i in range(j * j + j + 1,	(j + 2) * (j + 2),	j + 1   ):
      primes += is_prime(A_   )
    j += 2
  return j


if __name__ == "__main__":
    import doctest

    doctest.testmod()

